few_shots: Example:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Thought: This problem applies trigonometric identities, including the angle addition formula for cosine.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001894

HS
options: Analyze functions using different representations.

Construct and compare linear and exponential models and solve problems.

Define, evaluate, and compare functions.

Build new functions from existing functions.

Prove and apply trigonometric identities.

Build a function that models a relationship between two quantities.

Use functions to model relationships between quantities.

Extend the domain of trigonometric functions using the unit circle.

Model periodic phenomena with trigonometric functions.

Understand the concept of a function and use function notation.

Interpret functions that arise in applications in terms of the context.

Interpret expressions for functions in terms of the situation they model.
problem_activity: Activity
In this activity, students investigate how the height of water in a graduated cylinder is a function of the volume of water in the graduated cylinder. Students make predictions about how the graph will look and then test their prediction by filling the graduated cylinder with different amounts of water, gathering and graphing the data (MP4).
Digital
Launch
Arrange students in groups of 3–4. Be sure students know how to measure using a graduated cylinder. If needed, display a graduated cylinder filled to a specific measurement for all to see and demonstrate to students how to read the measurement. Give each group access to a graduated cylinder and water.
Give groups 8–10 minutes to work on the task, follow with a whole-class discussion.
For classrooms with access to the digital materials or those with no access to graduated cylinders, an applet is included here. Physical measurement tools and an active lab experience are preferred.
Speaking: MLR8 Discussion Supports.
Display sentence frames to support small-group discussion. For example, “I think ____ , because _____ .” or “I (agree/disagree) because ______ .”
Design Principle(s): Support sense-making; Optimize output for (explanation)
Student Facing
Use the applet to investigate the height of water in the cylinder as a function of the water volume.
Before you get started, make a prediction about the shape of the graph.
Check Reset and set the radius and height of the graduated cylinder to values you choose.
Let the cylinder fill with different amounts of water and record the data in the table.
Create a graph that shows the height of the water in the cylinder as a function of the water volume.
Choose a point on the graph and explain its meaning in the context of the situation.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 3–4. Be sure students know how to measure using a graduated cylinder. If needed, display a graduated cylinder filled to a specific measurement for all to see and demonstrate to students how to read the measurement. Give each group access to a graduated cylinder and water.
Give groups 8–10 minutes to work on the task, follow with a whole-class discussion.
For classrooms with access to the digital materials or those with no access to graduated cylinders, an applet is included here. Physical measurement tools and an active lab experience are preferred.
Speaking: MLR8 Discussion Supports.
Display sentence frames to support small-group discussion. For example, “I think ____ , because _____ .” or “I (agree/disagree) because ______ .”
Design Principle(s): Support sense-making; Optimize output for (explanation)
Student Facing
Your teacher will give you a graduated cylinder, water, and some other supplies. Your group will use these supplies to investigate the height of water in the cylinder as a function of the water volume.
Before you get started, make a prediction about the shape of the graph.
Fill the cylinder with different amounts of water and record the data in the table.
| volume (ml) |   |   |   |   |   |
|-------------|---|---|---|---|---|
| height (cm) |   |   |   |   |   |
Create a graph that shows the height of the water in the cylinder as a function of the water volume.
[IMG]
Choose a point on the graph and explain its meaning in the context of the situation.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Select groups to share the graph for the third question and display it for all to see. Consider asking students the following questions:
“What do you notice about the shape of your graph?”
“What is the independent variable of your graph? Dependent variable?”
“How does this graph differ from what you predicted the shape would be?”
“For the last question, what point did you choose, and what does that point mean in the context of this activity?”
“What would the endpoint of the graph be?” (There is a maximum possible volume for the cylinder. Once it’s filled, any extra water will spill out and not raise the water height.)
[IMG]
Ask students to predict how the graph would change if their cylinder had double the diameter. After a few responses, display this graph for all to see:Explain that each line represents the graph of a cylinder with a different radius. One cylinder has a radius of 1 cm, another has a radius of 2 cm, and another has a radius of 3 cm. Have students consider which line must represent which cylinder. Ask, “how did the slope of each graph change as the radius increased?” (As the radius is larger, the slope is less steep. This is because for a cylinder with a larger base, the same volume of water will not fill as high up the side of the cylinder.)


correct_option_index: 6